Parametric recursive digital filter

ABSTRACT

A digital filter includes a delay network with a plurality of delay elements configured and arranged as all-pass filters, having a controllable coefficient value. In the case of a low-pass or high pass filter, the cut-off frequency of the filter can be controlled via the controllable coefficient value associated with phase angle. Similarly, in a bandpass filter, the center frequency is set as a function of the controllable coefficient value.

CLAIM OF PRIORITY

[0001] This patent application claims priority to European PatentApplication serial number 02021903.6 filed on Sep. 27, 2002.

[0002] 1. Field of the Invention

[0003] This invention relates to the field of signal processing, and inparticular to a parametric recursive digital filter.

[0004] 2. Related Art

[0005] As known, digital filters include at least one delay unit and acoefficient network, which determine the cut-off/center frequency (i.e.,f_(c)) of the filter. In some application it is desirable to change thecut-off or centre frequency of a filter in response to a controlquantity (parametric filter).

[0006] In conventional digital filters, the cut-off or center frequencyis controlled with either the aid of coefficient look-up tables orreal-time calculation of the filter coefficients. However, knownstructures for implementing the coefficient look-up tables and/or thereal-time calculation require a rather large amount of memory and/orincreased computational ability. This is of course particularly true inthe case of higher-order filters. W. Schüssler, W. Winkelnkemper,suggest in the publication entitled “Variable Digital Filters”, Arch.Elektr. Übertr., Vol. 24, 1970, issue 11, pages 524-525, an arrangementin which the delay elements of a conventional digital filter arereplaced by all-pass filters in order to obtain a parametric filter.However, the authors of this publication concede that this procedurecannot be implemented in digital filters because the all-pass filterscomprise an attenuation when the highest all-pass filter coefficient isnot equal to zero. As a solution for this problem, it is suggestedeither to use only all-pass filters in which the highest all-pass filtercoefficient is equal to zero or only to use non-recursive filterstructures. A problem with the proposed solution is that few filters canbe implemented so that the possible applications are very limited.

[0007] Therefore, there is a need for a parametric recursive digitalfilter.

SUMMARY

[0008] The cut-off frequency or center frequency of a digital filter maybe controlled by changing a coefficient value within a delay unit of thefilter. A linear relationship exists between the coefficient value andthe cut-off or center frequency of the recursive digital filter.

[0009] The digital filter may be implemented with a recursive filterstructure, such as for example digital wave filter structures. A filterelement with controllable phase angle (e.g., all-pass filter ) isprovided as the delay unit.

[0010] In complex filters and higher-order filters, a plurality of delayunits may be used, and the delay units formed by identical filterelements selected in the same manner. As a result, the filter elementshave the same coefficient value so coefficient calculation only needs tobe performed once and can be used for all the filter elements.

[0011] In one embodiment, the delay unit may include an all-pass filter.The delay unit may include a first adder, one input of which forms theinput of the delay unit, and a second adder, the output of which formsthe output of the delay unit. A coefficient section is connected betweenthe output of the first adder and a first input of the second adder. Afirst delay element is connected between the input of the delay unit anda second input of the second adder, and a second delay element isconnected between the output of the delay unit and a second input of thefirst adder. The phase angle of the filter element may be adjusted bychanging the coefficient of the coefficient section, and the output ofthe first and/or second delay element being provided for connecting afeedback path. In addition to this all-pass filter embodiment, otherembodiments of an all-pass filter may also be used.

[0012] To simplify the structure of one embodiment of the all-passfilter, two delay units comprising delay elements can be interconnectedwith one another (in each case) in such a manner that only a total ofthree delay elements are provided, one delay element being used for bothdelay units.

[0013] Other systems, methods, features and advantages of the inventionwill be, or will become, apparent to one with skill in the art uponexamination of the following figures and detailed description. It isintended that all such additional systems, methods, features andadvantages be included within this description, be within the scope ofthe invention, and be protected by the following claims.

DESCRIPTION OF THE DRAWINGS

[0014] The invention can be better understood with reference to thefollowing drawings and description. The components in the figures arenot necessarily to scale, emphasis instead being placed uponillustrating the principles of the invention. Moreover, in the figures,like reference numerals designate corresponding parts throughout thedifferent views.

[0015]FIGS. 1A and 1B are block diagram illustrations of two first-orderall-pass filters;

[0016]FIG. 2 is a graph illustrating the phase response/frequencybending function, of a first-order all-pass filter as a function of theall-pass filter coefficient;

[0017]FIG. 3 is a graph illustrating the frequency resolution of aparametric filter for various all-pass filter parameters γ,

[0018]FIG. 4 is a block diagram illustration of a prior art finiteimpulse response (FIR) filter;

[0019]FIG. 5 is a block diagram illustration of a parametric finiteimpulse response filter;

[0020]FIG. 6 is a detailed block diagram illustration of acomputationally efficient structure for the parametric FIR filterillustrated in FIG. 5;

[0021]FIG. 7 is a block diagram illustration of a prior art biquadfilter as an example of a second-order infinite impulse response (IIR)filter;

[0022]FIG. 8 is a block diagram illustration of the structure of abiquad filter as an example of a second-order infinite impulse response(IIR) filter;

[0023]FIG. 9 is a block diagram illustration a parametric infiniteimpulse response filter with a low-pass filter as the first delay unit;

[0024]FIGS. 10A-10C illustrate a series of coordinate patternsillustrating the pole-zero shift in the Z domain as a function of theall-pass filter parameter γ in a second-order parametric IIR filterstructure;

[0025]FIG. 11 is a graph illustrating the pole-zero diagram of anormalized filter;

[0026]FIG. 12 is a graph illustrating the amplitude/frequency responseof the normalized filter;

[0027]FIG. 13 is a detailed block diagram illustration of acomputationally efficient embodiment of the second-order parametric IIRfilter illustrated in FIG. 9;

[0028]FIG. 14 is a graph illustrating the impulse response of a targetfilter for a second-order parametric IIR filter;

[0029]FIG. 15 is a graph illustrating the resultant amplitude/frequencyresponse of the target filter; and

[0030]FIG. 16 is a graph illustrating the pole-zero diagram of thetarget filter.

DETAILED DESCRIPTION

[0031] A digital filter includes delay elements configured and arrangedas all-pass filters (e.g., first-order low pass filters). The all-passfilter may be referred to as a frequency distorting filter since thespecifications of the filter are often implemented on a distortednon-uniform frequency axis.

[0032] A shift in the amplitude/frequency response |H(z)| of the filtercan be achieved if z is replaced or mapped by an expression W(z), whereW(z) meets the following conditions:

[0033] (i) the inside of the unit circle (Z plane) must be mapped againto the inside of the inner circle; and

[0034] (ii) the unit circle must be mapped to itself.

[0035] Condition (i) specifies that if H(z) is stable, H(W(z)) is alsostable. The second condition makes it possible to shift (i.e., “map”)the amplitude/frequency response into another arbitrary frequency range.

[0036] W(z) is the transfer function for an all-pass filter. Theamplitude/frequency response |H(z)| of the all-pass filter is constantlyequal to one, which ensures that the amplitude/frequency response of theoverall filter is not changed and that the first condition (i) issatisfied. Thus, a displacement or bending of the frequency axis occurs,and the displacement is controlled by the filter coefficient of thefirst-order all-pass filter. As a result, filter coefficient value γ ofthe first-order all-pass filter is also called frequency curvatureparameter.

[0037]FIGS. 1A and 1B are block diagram illustrations of first-orderall-pass filters. FIG. 1A illustrates a first order all-pass filter 20that receives an input signal sequence x[n] on a line 22. A summer 24receives the input signal on the line 22 and the past value of theoutput signal y[n−1] on line 26, and provides the resultant sum on line28. The sum is input to a coefficient section 30, which multiplies thesummed signal on the line 28 by coefficient value γ. The product isoutput on line 32 to summer 34, which also receives a signal on line 36indicative of the past value of the signal on the line 28. The summer 34provides output signal y[n] on line 38.

[0038]FIG. 1B illustrates another first order low pass filter 40. Thisfilter receives the input signal x[n] on line 42 and sums this signalwith a coefficient weighted feedback signal on line 44. A summer 46provides a summed value on line 48, which is input to a coefficientmultiplier 50 that multiplies the signal on the line 48 with coefficientvalue −γ, and a delay element 51. The resultant product is output online 52 and summed with a delayed version of the signal on the line 48,to provide output signal y[n] on line 54.

[0039] A first-order all-pass filter has the following transferfunction: $\begin{matrix}{{W(z)} = \frac{1 - {\gamma*z^{- 1}}}{z^{- 1} - \gamma}} & {{EQ}.\quad 1}\end{matrix}$

[0040] The first-order all-pass filter changes its phase response as afunction of the all-pass parameter γ, which can move within a range of−1<γ<+1. The change of this frequency response corresponds to bending ofthe frequency axis or mapping into a new frequency range.

[0041]FIG. 2 shows the frequency response/frequency bending function ofthe first-order all-pass filter as a function of the all-passcoefficient/frequency curvature parameter γ. Phase in degrees is plottedalong the vertical axis, while normalized frequency (i.e., frequencywith respect to the sampling frequency f_(s) divided by two) is plottedalong the horizontal axis. As shown, high frequencies are mapped to lowfrequencies for negative γ values, and low frequencies are mapped tohigh frequencies for positive γ values. The greater the amount of γ, thegreater the frequency bending. There is no frequency bending for thecoefficient value γ=0 —that is, the frequency axis is mapped to itselfagain in this case (bisector).

[0042] Since delay elements in traditional filter structures do notappear in the form of z but in the form of z⁻¹, it is also appropriatefor filters with frequency bending to work not with W(z), but withW⁻¹(z). In this way, it is impossible to replace all delay elements z⁻¹with their frequency-dependent counterpart: $\begin{matrix}{{W^{- 1}(z)} = {{D(z)} = {\frac{z^{- 1} - \gamma}{1 - {\gamma*z^{- 1}}}.}}} & {{EQ}.\quad 2}\end{matrix}$

[0043] Due to the bending of the frequency axis, some frequency rangesare resolved more finely and others more coarsely in dependence on thefrequency curvature parameter γ. If a filter has a local frequencyresolution of Δf(f) before the “bending”, the bent filter has a localfrequency resolution of Δf_(w)(f), where the following holds true:$\begin{matrix}{{\Delta \quad {f_{w}(f)}} = {\Delta \quad {f( {f_{w}( {f,\gamma} )} )}*\frac{1 - \gamma^{2}}{1 + \gamma^{2} + {2*\gamma*{\cos ( \frac{2*\pi*{f_{w}( {f,\gamma} )}}{f_{S}} )}}}}} & {{EQ}.\quad 3}\end{matrix}$

[0044] where f_(s) is the sampling frequency and f_(w)(f, γ) stands fora function that specifies the corresponding frequency in the bentfrequency range for a given frequency and all-pass parameter. Thefrequency of the parametric filter as a function frequency f and theall-pass parameter γ can be expressed as: $\begin{matrix}{{f_{w}( {f,\gamma} )} = {f + {\frac{f_{S}}{\pi}*{\arctan ( \frac{\gamma*{\sin ( \frac{2*\pi*f}{f_{S}} )}}{1 - {\gamma*{\cos ( \frac{2*\pi*f}{f_{S}} )}}} )}}}} & {{EQ}.\quad 4}\end{matrix}$

[0045] The turning point frequency f_(tp) is the frequency that thefrequency resolution of the parametric filter is the same as before thefrequency bending. It can be shown that the following holds true forf_(tp): $\begin{matrix}{f_{tp} = {\frac{f_{S}}{2*\pi}*{\arccos (\gamma)}}} & {{EQ}.\quad 5}\end{matrix}$

[0046] For frequencies less than f_(tp), the frequency resolution isimproved with a positive γ. The frequency resolution decreases fornegative γ. FIG. 3 illustrates the frequency resolution of theparametric filter for various all-pass parameters γ. The turning pointfrequency f_(tp) is given when the frequency resolution is equal to one.

[0047]FIG. 4 is a block diagram illustration of a prior art finiteimpulse response (FIR) filter. The filter includes delay elements 62,64. A frequency-bending FIR filter is obtained by replacing the delayelements 62, 64 (FIG. 4) with frequency-dependent delay elements 66, 68having a transfer function D(z), as shown in FIG. 5.

[0048]FIG. 5 is a block diagram illustration of a parametric finiteimpulse response (FIR) filter 65. If the frequency-dependent delaysections 66, 68 are replaced by their corresponding all-pass filterstructure (e.g., by the structure shown in FIG. 1A), and the structureis simplified by dissolving redundant branches, a computationallyefficient structure of a frequency-bending FIR filter is obtained asillustrated in FIG. 6.

[0049]FIG. 6 is a detailed block diagram illustration of acomputationally efficient embodiment for the parametric FIR filterillustrated in FIG. 5. The filter includes a plurality ofseries-connected delay elements (e.g., 70-72), summing nodes 74,75 eachinserted between two delay elements, and coefficient sections 78, 80.The input signals for the coefficient sections 78, 80 are picked up atthe input of the delay element 70, the output of the summing node 74,and output of delays 71, 72. The coefficient sections 78, 80 includeprogrammable/adjustable coefficient value γ. The coefficient sections78, 80 provide coefficient section output signals on line 82, 84,respectively, which are summed with associated delay element outputsignals to provide output signals to a multiplication network 90.Products from the multiplication network 90 are input to a summer 92,which provides an output signal on line 94.

[0050] Replacing a delay element in one of the traditional IIR filterstructures with a frequency-dependant (dispersive) delay element as hasalready been explained for the FIR filter provides a structure thatcontains feedback branches are free of delay, but which cannot beimplemented in this way, as has been explained in the prior artmentioned initially. FIGS. 7 and 8 show a recursive biquad filter beforeand after the mapping, respectively. FIG. 7 is a block diagramillustration of a prior art biquad filter as an example of asecond-order infinite impulse response (IIR) filter. FIG. 8 is a blockdiagram illustration of a modified biquad filter as an example of asecond-order infinite impulse response (IIR) filter. The structure of afrequency-bending IIR filter (i.e., frequency warping IIR filter—WIIR)is obtained by replacing the delay elements illustrated in FIG. 7 withfrequency-dependent delay elements 80, 82 having a transfer functionD(z), as shown in FIG. 8. FIG. 7 is based on the arrangement illustratedin FIG. 4, and has been expanded by a feedback network.

[0051]FIG. 8 illustrates a WIIR filter that is suitable for higher orderfilters, and provides for all other delay elements apart from the firstone to be replaced by first-order all-pass filters D(z). The first delayis replaced by a first-order low-pass filter H(z). This measure makes itnecessary to recalculate (to map) the filter coefficients of theoriginal IIR filter to the new structure, which is done by a relativelysimple recursive formula. FIG. 9 shows the corresponding frequencymapping/warping IIR filter structure that includes a low-pass filterH(z) 100 and a plurality of first-order all-pass filters D(z) 102, 104.

[0052] The digital filters according to an aspect of the presentinvention include delay elements configured and arranged as all-passfilters (e.g., particularly of the first order). However, to avoid thezero-delay feedbacks, feedback is not effected after the respectivedispersive delay units (in the prior art) but, according to theinvention, directly after the delay element within a dispersive delayunit. The forward branch (FIR part of the filter) with its coefficientsb₀, b₁, b₂ etc. remains unaffected by this measure. We shall now discussa procedure for designing a second-order frequency-bending IIR filterwith the design of a 10 Hz high-pass filter.

[0053] Using frequency-bending filters, a fixed “normalized filter” canbe designed that has an arbitrary fixed cut-off frequency f_(c), andwhich can be displaced into any desired frequency range with the aid ofthe frequency curvature parameter γ. FIGS. 10A-10C illustrate thepole-zero shift in the Z domain as a function of the frequency curvatureparameter γ.

[0054] Since originally filters are assumed that are designed in thedirect form, it is appropriate to utilize the advantages of the directform as far as possible. The frequency range within which the directform has particularly good characteristics in the sense of quantizationsensitivity and dynamic range, is around f_(s)/4. For this reason, thecut-off frequency f_(c) of the prototype filter is left atf_(c)=f_(s)/4, where f_(s) is the sampling frequency. FIGS. 11 and 12show the associated pole-zero diagram and the amplitude/frequencyresponse of the normalized filter, respectively.

[0055] The frequency curvature parameter γ is be calculated so thenormalized filter is shifted towards the required cut-off frequency. Forthis purpose, the equation: $\begin{matrix}{{f_{pre}( {f,\gamma} )} = {f + {\frac{f_{S}}{\pi}*{\arctan ( \frac{\gamma*{\sin ( \frac{2*\pi*f}{f_{S}} )}}{1 - {\gamma*{\cos ( \frac{2*\pi*f}{f_{S}} )}}} )}}}} & {{EQ}.\quad 6}\end{matrix}$

[0056] can be transformed in such a manner that the frequency curvatureparameter γ can be calculated from it. The calculation formula thatproduces the frequency curvature parameter γ with the aid of the fixedcut-off frequency of the normalized filter f_(pre) and the desiredcut-off frequency f_(c) can be expressed as: $\begin{matrix}{\gamma = {- ( \frac{\tan ( \frac{( {f_{pre} - f_{c}} )*\pi}{f_{s}} )}{{\sin ( \frac{2*\pi*f_{c}}{f_{s}} )} + {{\cos ( \frac{2*\pi*f_{c}}{f_{s}} )}*{\tan ( \frac{( {f_{pre} - f_{c}} )*\pi}{f_{s}} )}}} )}} & {{EQ}.\quad 7}\end{matrix}$

[0057] If f_(pre) is left at f_(s)/4, the previous equation is reducedto: $\begin{matrix}{\gamma = {\tan ( {\frac{\pi}{4} - \frac{f_{c}*\pi}{f_{s}}} )}} & {{EQ}.\quad 8}\end{matrix}$

[0058] The expression f_(pre) for the fixed cut-off frequency of thenormalized filter was selected since it can be said that this filter isdesigned in the distorted frequency range. One can return from thedistorted frequency range into the original frequency range by operatingthe parametric filter with the frequency curvature parameters γcalculated above.

[0059]FIG. 13 illustrates a computationally efficient embodiment of asecond-order parametric IIR filter. This embodiment is obtained byexpanding the arrangement according to FIG. 6 by one feedback network.From the second-order parametric IIR filter, the impulse response shownin FIG. 14 and the resultant amplitude/frequency response of the targetfilter shown in FIG. 15 is obtained. The associated pole-zero diagram ofthe target filter is shown in FIG. 16.

[0060] The digital filters may be implemented in microprocessors, signalprocessors, microcontrollers, computing devices et cetera. Theindividual filter components such as, for example, delay units, delayelements, coefficient sections are then hardware components of themicroprocessors, signal processors, microcontrollers, computing devices,et cetera that are correspondingly used by the executable software.

[0061] The illustrations have been discussed with reference tofunctional blocks identified as modules and components that are notintended to represent discrete structures and may be combined or furthersub-divided. In addition, while various embodiments of the inventionhave been described, it will be apparent to those of ordinary skill inthe art that other embodiments and implementations are possible that arewithin the scope of this invention. Accordingly, the invention is notrestricted except in light of the attached claims and their equivalents.

What is claimed is:
 1. A parametric recursive digital filter having acut-off/center frequency, said digital filter comprising: a delay unithaving a delay element and an interconnected phase network that includesan controllable phase angle, where the cut-off/center frequency of saiddigital filter is set as a function of said controllable phase angle; apositive feedback network connected to said delay unit creating apositive feedback path; and a feedback network connected to said delayunit creating a feedback path connected to the output of the delayelement in the delay unit.
 2. The filter according to claim 1, in whicha plurality of delay units are provided.
 3. The filter according toclaim 2, in which the delay units are identically designed and arecontrolled in the same manner.
 4. The filter according to claim 1,wherein the delay unit comprises a delay element.
 5. The filteraccording to claim 1, wherein the positive feedback network comprises aplurality of positive feedback paths.
 6. The filter according to claim1, wherein the feedback network comprises a plurality of feedback paths.7. The filter according to claim 1, wherein said delay unit comprises anall-pass filter.
 8. The filter according to claim 7, wherein saidall-pass filter comprises: a first adder, one input of which forms theinput of the delay unit, a second adder, the output of which forms theoutput of the delay unit, a coefficient section which is connectedbetween the output of the first adder and a first input of the secondadder, a first delay element which is connected between the input of thedelay unit and a second input of the second adder, a second delayelement which is connected between the output of the delay unit and asecond input of the first adder, the phase angle of the filter elementbeing adjustable by changing the coefficient of the coefficient section,and the output of the first and/or second delay element being providedfor connecting a feedback path.
 9. The filter according to claim 9, inwhich two delay units comprising delay elements are interconnected withone another in such a manner that only a total of three delay elementsare provided, one delay element being attributable to both delay units.10. The filter according to claim 6, in which a frequency-influencingfilter unit is provided as delay unit.
 11. A digital filter thatreceives an input signal, comprising: a delay network that receives saidinput signal and provides a delay network output signal, and comprises atime delay element configured and arranged as an all-pass-filter havinga programmable coefficient value γ, a multiplication network thatreceives and multiplies said input signal and said delay network outputsignal by uniquely associated coefficient values to provide a weightedinput signal and a weighted delay network output signal; and a summingnetwork that receives and sums said weighted input signal and saidweighted delay network output signal to provide a filter output signal.